Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring

We compute the conservative piece of the gravitational self-force (GSF) acting on a particle of mass m(1) as it moves along an (unstable) circular geodesic orbit between the innermost stable orbit and the light ring of a Schwarzschild black hole of mass m(2) >> m(1). More precisely, we construct the function h(uu)(R,L)(x) equivalent to h(mu v)(R,L)u(mu)u(v) (related to Detweiler's gauge-invariant "redshift" variable), where h(mu v)(R,L)(proportional to m(1)) is the regularized metric perturbation in the Lorenz gauge, u(mu) is the four-velocity of m(1) in the background Schwarzschild metric of m(2), and x equivalent to [Gc(-3)(m(1) + m(2))Omega](2/3) is an invariant coordinate constructed from the orbital frequency Omega. In particular, we explore the behavior of h(uu)(R,L) just outside the "light ring" at x 1/3 (i.e., r = 3Gm(2)/c(2)), where the circular orbit becomes null. Using the recently discovered link between h(uu)(R,L) and the piece a(u), linear in the symmetric mass ratio v equivalent to m(1)m(2)/(m(1) + m(2))(2), of the main radial potential A(u, v) = 1 - 2u + va(u) + O(v(2)) of the effective-one-body (EOB) formalism, we compute from our GSF data the EOB function a(u) over the entire domain 0 < u < 1/3 (thereby extending previous results limited to u <= 1/5). We find that a(u) diverges like a(u) approximate to 0.25(1 - 3u)(-1/2) at the light-ring limit, u -> (1/3)(-), explain the physical origin of this divergent behavior, and discuss its consequences for the EOB formalism. We construct accurate global analytic fits for a(u), valid on the entire domain 0 < u < 1/3 (and possibly beyond), and give accurate numerical estimates of the values of a(u) and its first three derivatives at the innermost stable circular orbit u = 1/6, as well as the associated O(v) shift in the frequency of that orbit. In previous work we used GSF data on slightly eccentric orbits to compute a certain linear combination of a(u) and its first two derivatives, involving also the O(v) piece of a second EOB radial potential (D) over bar (u) = 1 + v (d) over bar (u) + O(v(2)). Combining these results with our present global analytic representation of a(u), we numerically compute (d) over baru on the interval 0 < u <= 1/6.